GEOS 36501/EVOL 33001 27 January 2012 Page 1VII-VIII. Some case studies using branchingmodels1 Examples of forward problems1.1 Stochastic survivorship of single clade1.1.1 Relevant equations: A11-A14 from Raup 1985.1.1.2 Example: waning of tril obi tes from Cambrian through Permian(Raup, 1981, Acta Geologica Hispanica 16:25-33).• Could trilobites have stochastically drifted to extinction (“Galton extinction”) withhigh initial diversity if p = q?• Look for conditions that yield intermediate probability of clade extinction (too low,and it is unlikely to have happened; too high, and trilobites should have becomeextinct much sooner, and most other clades should also have become extinct).• If not, how much higher would q have to be to account for drift to extinction?• Note importance of sensitivity analysis: explore wide range of parameters, not justyour “best guess”.GEOS 36501/EVOL 33001 27 January 2012 Page 2GEOS 36501/EVOL 33001 27 January 2012 Page 3GEOS 36501/EVOL 33001 27 January 2012 Page 41.2 Stochastic survivorship of multiple clades1.2.1 Relevant equations: A11-A14 from Raup 1985, plus modifications (seeappended text).1.2.2 Example: thought experiment on multliple origins of life (Raup andValent ine , 1983, PNAS 80:2981-2984).• Look for conditions that yield intermediate probability of bioclade survival. (Too lowand it’s unlikely that life would have survived at all; too high and Phanerozoic lifeshould be polyphyletic.)• Again note importance of exploring range of parameters.From Raup and Valentine (1983)GEOS 36501/EVOL 33001 27 January 2012 Page 5From Raup and Valentine (1983)GEOS 36501/EVOL 33001 27 January 2012 Page 61.3 Expected longevity of clades with variat io n in starti ngrichness reflecting temporal variation in species-level rates1.3.1 Relevant equations: A7-A10 and A15-A18 from Raup 19851.3.2 Increased longevities of genera that originate right after massextinctions; may this result from high speciation rate in post-extinctionwo rl d (Miller and Foote, 2003, Science 302:1030-1032)?1.3.3 If specation rat e higher ri ght after mass e xtinct io n, then gener a willaccumul ate more speci es and will persist longer even if speciation andspecies extinction rates are subsequently equal.Fig. 1. Mean longevi-ties of genus cohortsoriginating throughoutthe Phanerozoic andlatest Neoproterozoic(solid lines). (A) Maxi-ma are labeled; statisti-cal significance is indi-cated when the peakexceeds the upper 95%confidence interval(dotted line) derivedwith a randomizationprocedure (15). (B)Mean genus longevitiescompared with percapita extinction rates(dotted line). Extinctionpeaks that precede lon-gevity maxima are la-beled. (C) The patternafter culling generathat became extinctduring several post-Paleozoic mass extinc-tions (18). (D) The pat-tern after culling generathat survived to thepresent day; dotted lineas in (A) but based onlyon genera that did notsurvive to the presentday. Cm, Cambrian; O,Ordovician; S, Silurian; D,Devonian; C, Carbonifer-ous; P, Permian; Tr, Tri-assic; J, Jurassic; K, Creta-ceous; T, Tertiary.encemag.org SCIENCE VOL 302 7 NOVEMBER 2003 1031GEOS 36501/EVOL 33001 27 January 2012 Page 7 Miller and Foote-3Equations for Birth-Death ModelThe probability that a genus will have n species after an elapsed time t, assuming that n >0 and p > q, is given by p(n ,t) = (1-B)Bn-1, where B = A (p/q) and}])[(1}/{])[({ qtqpexpptqpexpqA −−⋅−−⋅= , and p and q are the per-capitaorigination and extinction rates. This approach tacitly assumes that the genera inquestion originate at the start of this initial interval. Given an initial standing diversity ofn, and assuming that speciation and species-extinction rate are equal, the median genusduration is equal to)/ 1)/(21()(1/−=nmpnT . Therefore, the expected medianduration for a genus, assuming that it originates at the beginning of the initial timeinterval and is still extant at the end of it, is equal to ¦∞=⋅=1),()(ntnpnmTmT .We focus in this analysis on the median duration because it is analytically more tractablethan the mean, but similar reasoning holds for mean durations. For compatibility withour empirical results, only genera that survive to the end of their initial substages areconsidered.Fig. 2. Elevation in median genus duration pro-duced during an interval in which speciation rate( p) within the genus exceeds species extinctionrate (q). The initial interval is assumed to be 4 My,about the length of a substage in our analysis.GEOS 36501/EVOL 33001 27 January 2012 Page 81.4 Early origins of major biologic groups (Raup, 1983,Paleobiology 9:107-115) ( See rel evant pages from this paper,reproduced below.)1.4.1 Branchi ng structure o f evolutionary trees predicts many extinct taxa,and just a few living ones; the living ones have deep roots.1.4.2 Thus, the homog enous branching model predicts that randomly chosen,living species are likely to have a divergence time deep in the past.From Raup (1983)GEOS 36501/EVOL 33001 27 January 2012 Page 9GEOS 36501/EVOL 33001 27 January 2012 Page 10GEOS 36501/EVOL 33001 27 January 2012 Page 11GEOS 36501/EVOL 33001 27 January 2012 Page 12From Principles of Paleontology 3/e1.4.3 Within- vs. between-group pairs• Let there be g groups with Nispecies, i = 1, ..., g.• Within-group pairs:Pgi=1Ni(Ni− 1)/2.• Between-group pairs:Pi,j(i6=j)NiNj.1.4.4 Sensitivity• Lower rates lead to deeper divergence.• Higher net diversification rate leads to shallower divergences.• Thus, single large group with recent radiation (e.g. insects) can greatly skewempirical results; this would represent clear deviation from homogeneous model.GEOS 36501/EVOL 33001 27 January 2012 Page 132 Further inverse problem: Estimating species-level pand q from genus-level survivorship2.1 Assume genus survivorship depends only on species-levelrates and elapsed time—i.e. there is no distinct genus-levelextinction process.2.2 We can infer something about a process below our scale ofresolution based on observations at a coarser scale.2.3 This ability comes at a cost: the assumption oftime-homogeneous, taxonomically homogeneous model.2.4 For candidate values of p and q, predict e xpected genus-levelsurvivorship curves. Compare observed data to expectedcurve and find the values of p and q that maximize fit(whether by likelihood or some other criterion).2.4.1 Let tibe time si nce the orig in of the cohort (by whatever convention isused to pl ace the